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Abstract

Background

The forces which affect homelessness are complex and often interactive in nature.
Social forces such as addictions, family breakdown, and mental illness are compounded
by structural forces such as lack of available low-cost housing, poor economic conditions,
and insufficient mental health services. Together these factors impact levels of homelessness
through their dynamic relations. Historic models, which are static in nature, have
only been marginally successful in capturing these relationships.

Methods

Fuzzy Logic (FL) and fuzzy cognitive maps (FCMs) are particularly suited to the modeling
of complex social problems, such as homelessness, due to their inherent ability to
model intricate, interactive systems often described in vague conceptual terms and
then organize them into a specific, concrete form (i.e., the FCM) which can be readily
understood by social scientists and others. Using FL we converted information, taken
from recently published, peer reviewed articles, for a select group of factors related
to homelessness and then calculated the strength of influence (weights) for pairs
of factors. We then used these weighted relationships in a FCM to test the effects
of increasing or decreasing individual or groups of factors. Results of these trials
were explainable according to current empirical knowledge related to homelessness.

Results

Prior graphic maps of homelessness have been of limited use due to the dynamic nature
of the concepts related to homelessness. The FCM technique captures greater degrees
of dynamism and complexity than static models, allowing relevant concepts to be manipulated
and interacted. This, in turn, allows for a much more realistic picture of homelessness.
Through network analysis of the FCM we determined that Education exerts the greatest force in the model and hence impacts the dynamism and complexity
of a social problem such as homelessness.

Conclusions

The FCM built to model the complex social system of homelessness reasonably represented
reality for the sample scenarios created. This confirmed that the model worked and
that a search of peer reviewed, academic literature is a reasonable foundation upon
which to build the model. Further, it was determined that the direction and strengths
of relationships between concepts included in this map are a reasonable approximation
of their action in reality. However, dynamic models are not without their limitations
and must be acknowledged as inherently exploratory.

Keywords:

Background

Homelessness

Homelessness is a complex social problem with a variety of underlying economic and
social factors such as poverty, lack of affordable housing, uncertain physical and
mental health, addictions, and community and family breakdown. These factors, in varying
combinations, contribute to duration, frequency, and type of homelessness. To be fully
homeless is to live without shelter; however, many experience partial homelessness
that can include uncertain, temporary, or sub-standard shelter. Homelessness is difficult
to define, thus governments struggle with uncertainty when creating and implementing
policies they hope will effectively manage or eradicate this problem.

Levels of government, in countries like Canada, add to the complexity of dealing with
homelessness. Being governed at three different levels, federal, provincial, and municipal,
requires high levels of agreement to effectively create and administer policies. In
Canada, each level of government is responsible for different facets of homelessness.
The federal government, responsible for the whole of Canada, creates and administers
policies and funding for aboriginal peoples (a segment of Canada’s population over-represented
in homeless counts), seniors, and social housing, as well as transfers funds to the
provinces to help pay for their social programs. The provincial government, responsible
for needs of the provinces and territories, creates and administers policies regarding
mental illness, addictions, welfare, minimum wage laws, landlord and tenant acts,
and child protection services and shares responsibility with the federal government
for seniors and social housing. The municipal governments, are seen as the hands or
arms of the provincial government, and are technically not responsible for homelessness;
however are often involved in choosing sites for social housing, supporting emergency
shelters and hospital emergency wards, as well as providing support, in a variety
of ways, to facilitate these initiatives. The fact that there is no comprehensive
national housing strategy to co-ordinate these levels of government often leads to
inadequate policies and funding that fall far short of meeting the country’s housing
needs [1]. This lack of coordination towards policy and funding for homelessness has recently
come to the attention of courts in Canada who have begun to make decisions which support
shelter as an essential right for Canadians [2]. The UN Special Rapporteur on adequate housing in Canada has also strongly urged
the federal government to commit sufficient funding to create a national housing strategy
by working with the provinces and territories [3].

Metro Vancouver is one city in Canada which conducts a comprehensive homeless count
every three years [4]. Counters make every effort to include in the count those considered sheltered homeless
(individuals who spend nights in shelters, safe houses, transition houses, hospitals,
jails, remand centres, and detox/recovery facilities) and those who are unsheltered
homeless (individuals who spend their nights unsheltered on streets, in parks, or
at drop-in programs). Counts are shown in Figure 1.

It becomes apparent that if the complex and oft-times chaotic experiences such as
job loss that lead to family breakdown, mental illness, and drug/alcohol addiction,
which may lead to homelessness, were better understood then social policies and procedures
which constitute “best practices” would be more effective in reducing and preventing
homelessness [5]. Fuzzy logic and fuzzy cognitive maps are especially useful for modelling complex
social problems due to their inherent ability to capture and model vague concepts
and values [6]. In relationship to homelessness, syllogisms such as, “if there is a lack of affordable
housing, then there will be a significant increase in homelessness” can be accurately
modelled by assigning a value to the parameter based on the retrieved linguistic terms
taken from existing empirical literature. In this way greater meaning, which captures
and aggregates the nuances of the stressors and protective factors, is given to the
existing empirical literature related to homelessness. This also allows the complex
social issue to be graphically described in a manner which may be more readily understood.
This, in turn, may then help social policy-makers to refine their decision-making,
leading to effective changes in social policies with the goal of reducing homelessness.

Fuzzy logic (FL) is a mutli-valued logic technique that is approximate. Rather than
using traditional logic theory where binary sets have a two-valued logic (i.e., true,
1, and false, 0), fuzzy variables have a truth-value between 0 and 1, allowing them
to be valued between absolutely true and absolutely false. Using linguistic variables,
taken from empirical literature that describes the effect each factor in a knowledge
system has on the others, FL can be used to convert the effects into values between
0 and 1. Once determined, these values can then be input into a graphical representation
of the system containing all factors with directed lines (edges) showing the calculated
strength of the causal relationship between them. This graphical representation is
known as a fuzzy cognitive map (FCM). A brief description of the techniques, with
an example is presented in the subsequent subsection.

Fuzzy Cognitive Map (FCM)

The FCM is a framework used for modelling interdependence between concepts in the
real-world [7]. This is achieved by graphically representing the causal reasoning relationships
between vague or un-crisp concepts [6,8]. FCMs allow scientists to construct virtual worlds in which some of the complex and
interdependent concepts of a scenario can be captured and their interactions or causal
relationships modelled. Knowledge representation in these maps has an acquisition-processing
trade-off. FCMs, by providing a fuzzy graph structure for systematic causal propagation
and ease in processing fuzzy knowledge, are applicable in soft-knowledge domains such
as the social sciences. At the core of the FCM structure are the concepts to be studied
and modelled. Concepts can be understood to represent actors or the parts of the environment
which have impact on some phenomenon of interest (and each other), such as those included
in the simple FCM of heart disease illustrated in Figure 2.

Concepts which have no impact on other concepts are not represented via links on the
map, however are represented in the subsequent constructed adjacency matrix W and denoted, 0.

As can be seen in Figure 2, there is no direct effect of BW on C and therefore no link is drawn between these two concepts. The weight values {−1,0,1}
are used at this stage for simplicity and testing the FCM and are later refined through
the application of empirical linguistic terms and modifiers processed through FL.

The use of an FCM is particularly advantageous for graphically representing the interacting
relationships of concepts which appear in phenomena related to social science, political
science, organizational theory, military science, and international relations [8]. The connection matrix, W, may also be defined algebraically, demonstrating the influence concepts have on
one another [7].

Let us denote the ith concepts of a system as Ci. Then the value Ai, of a concept Ci, expresses the quantity of its corresponding physical value. The FCM converges to
a steady state when:

(1)

At each step, the value Ai of a concept is influenced by the values of concepts-nodes connected to it and is
updated according to the following formula:

(2)

where Ai(k) is the value of concept Ci at step k, Aj(k) is the value of concept Cj at step k, Wji is the weight of the interconnection from concept Cj to concept Ci and f is the threshold function used to bound the transformation to a limit cycle. In this
example, f(x) is a sign function defined in MATLAB [9] with the following functionality:

(3)

Following our heart disease example, consider: the concept, E, is active for some
individual. Therefore, E=1. No information is available for all other concepts in the map. Therefore, FH=0, C=0, BW=0, and BP=0. This is expressed by a vector C1 = (1,0,0,0,0,0). According to equations 2 and 3, the processing is listed in Table
1.

The right arrow indicates the threshold function operation in Equation 3. The above
results demonstrate that it takes four steps for the system to converge to a stable
state (limit cycle). The vector C4 demonstrates that the increase in E eventually leads to decreases in C, BW, BP, and
HD.

The FCM created for our study provides a graphical description of homelessness and
facilitates increased understanding of this complex social problem. Through simulation,
the usefulness of such a model is demonstrated and implications for its use in policy
decision-making are explored. As shown, FCMs related to complex social problems, allow
for refinement of knowledge through graphical understanding and simulations that may
be useful in improving social policies with the goal of reducing homelessness.

Methods

Virtual common-sense map of homelessness

First a virtual common-sense map was built based on the researchers’ personal and
historical knowledge of the factors which they perceived to affect homelessness. Using
homelessness as the central hub of the map, concepts which directly or indirectly,
positively or negatively affected homelessness, and each other, were linked through
directed edges. Each edge was assigned a weight depending on whether the antecedent
concept exerted a positive effect (+1) or a negative effect (−1) on the consequent
concept (Figure 3). Three prototypical cases were then developed and the model was run to ensure it
would function in accordance with the determined relationships prior to the actual
weights on the edges being refined through a literature search for the linguistic
terms.

Experimentation: Virtual common-sense map

Experimentation with the virtual common-sense model was conducted to ensure that it
would perform as expected and reach a stable state after iterating prior to the input
of the actual weight values. Sample cases were constructed with the goal of describing
an extreme case, most likely to result in homelessness; an extreme case, least likely
to result in homelessness; and a middle case, more closely representing the possibilities
of the real world, in which the likelihood of homelessness would be uncertain, see
Table 2.

•Case 1: In this scenario, the protective factor of rental subsidy was incapable of preventing
the negative social factors, criminal justice system involvement, addictions, and
mental illness from overwhelming the model - resulting in certain homelessness.

•Case 2: In this scenario, the protective factors of education and increased income resulted
in the elimination of the need for non-government assistance and a decrease in the
likelihood of criminal justice system interaction. This is a highly likely outcome
given that those with higher incomes and education are better able to identify and
seek help for their mental illnesses which increases the likelihood that they will
avoid incarceration. However, the strength of income and education as protective factors
against increasing mental illness is shown to be ineffective and the level of mental
illness continues to rise. Despite the increase in mental illness, education and income
will ensure an ongoing ability to provide shelter, resulting in homelessness being
an extremely unlikely outcome.

•Case 3: In this scenario, at the end of iteration 1, the effects of addiction, prior criminal
justice system involvement, and family breakdown are held at bay by the protective
factors of income, education and counselling. However, due to the known cumulative
negative effects of addiction, social isolation increases, signalling the likelihood
that, over time, there will be an increased possibility of family breakdown and greater
challenges controlling the addiction resulting in the increased likelihood of crime.
Iteration 2 demonstrates the actions of all the concepts present in iteration 1 continuing
to exert force on the model with the addition of an increase in mental illness caused
by the ongoing addiction resulting in an increasing likelihood of homelessness. As
the model continues to iterate, the addictions contribute to increasing social isolation
and criminal behavior resulting in a greater likelihood of family breakdown. At this
point the protective factors of education, income and counselling are overwhelmed
by the ongoing addictions and resulting mental illness and crime and the likelihood
of homelessness rises. However, given that education and income continue to exert
force, homelessness is not a certainty.

Given the fully explainable results of the model and the fact that it was able to
achieve stability after iterating, it was determined that the model functioned properly,
and the process of refining the concepts through the search of timely empirical literature
was conducted.

Fuzzy Cognitive Map of homelessness supported by empirical studies

To refine the edge weights on the FCM, timely, empirical literature was searched.
The original causal map was referred to for the paired concepts such as, education
and homelessness. These linked terms were then searched using the academic search
engine, Google Scholar. Numerous articles were retrieved and scanned for each pair
of linked concepts using only recently published (since the year 2000), peer reviewed,
empirical articles. This culminated in the capture of three linguistic statements
per concept pair for use in refining the map (see Table 3). Linguistic statements were required to be in the antecedent - consequent form as
earlier described. In the process of searching, paired concepts were refined (edges
and concepts added and removed from the virtual common-sense map Figure 3 after though deliberation with research team) resulting in a final map of 14 concepts
and 31 edges (Figure 4). To maintain the semantic consistency amongst various concepts, Oxford Canadian
Dictionary [10] was followed.

To calculate the quantitative weight values for each edge, first the qualitative weight
values for each of the retrieved linguistic terms was assessed. A Likert-type scale
was devised to determine the qualitative weight of each linguistic term. The values,
Very Low (VL), Low (L), Medium (M), High (H), and Very High (VH) were used to categorize
each term. We only consider five qualitative values for the sake of simplicity. However,
the scale could be less or more than five, depending on the intricacies of the system
under consideration. Consensus on meaning was achieved through discussion and vote.
This process resulted in a scale of ordered and ranked values for each concept pair.
For example, it might be stated in one peer-reviewed study that the effect of concept
A on concept B was, “profound”; whereas another article may state that the effect
was, “significant”. These statements, “profound” and “significant”, would be then
ranked on the Likert- type scale in reference to their absolute meaning as well as
their relative meaning. Thus, “profound”, would be valued as VH and “significant”
would be valued as H. In the case of disagreement or uncertainty regarding the precise
meaning of the words, Oxford Dictionary Online was referenced for definitions and
synonyms. A word bank was constructed during this process listing all the retrieved
terms for both comparative reference and to ensure consistency in the rankings, see
Table 4. Once the different qualitative weight values were determined for each linguistic
term, they were then collected into their groups of three and applied to the revised
FCM.

Subsequent to the information from the literature review having been transferred to
the FCM, the resulting map contained the concepts, the antecedent - consequent relationships
indicated via edges, the weight value of each edge (five qualitative, linguistic terms
- VL, L, M, H, VH), and the sign value showing the type of the influence (+ or −).
Following the application of the qualitative values to the FCM the values were then
converted to quantitative weight values using FL theory. Each link was first expressed
as a fuzzy rule then used in the Fuzzy Inference System (FIS) to generate a crisp
numeric value. For example, if the linguistic term retrieved from the literature was:
“The impact of concept A is profound on concept B”. It would then be converted to:
“The impact of concept A is VH on concept B”. This graded statement would then be
transformed using the rule statement:

The linguistic term ON is a binary variable. VH is defined using the triangular fuzzy
membership function, as shown in Figure 5. ON denotes the presence of the concept and VH denotes the weight value (qualitatively).
For simplicity sake, triangular membership functions have been used as suggested in
[85]. Interested readers can find more detailed explanation on membership functions in
[86].

•Example 1: As explained in the previous section, all qualitative values assigned to the edges
came from the literature review. As shown in Figure 6, “addiction” has a positive impact on homelessness. This means that an increase in
addiction in a society will lead to an increase in levels of homelessness. The three
linguistic terms related to “addiction”, extracted from the literature, were converted
to the fuzzy notion of rules as follows:

•The degree of impact was then converted from its qualitative value (M, H, VH) to
its quantitative value of 0.648 using FL concepts as described in [87]. All three studies indicated that as levels of addiction increase they exert a positive
effect resulting in increases in levels of homelessness. Therefore, it can be stated
that addictions affect homelessness by a factor of +0.648.

•Example 2: As shown in Figure 7, education has a negative effect on homelessness. This means that with higher levels
of education in a society there will be lower levels of homelessness. Therefore, the
impact of education on homelessness is modeled as negative - increases in education
lead to decreases in homelessness. All literature scanned indicates that as education
rises, homelessness falls. The first study stated that the impact of education on
homelessness was low, the second, medium, and the third, high. This information is captured to construct a rule base for a Fuzzy Inference System
(FIS). For each edge, we constructed an individual FIS and the defuzzified value,
in this case 0.5, is assigned to the edge. More information about the procedure can
be found in [87-90].

•Similarly, each edge was given a quantitative weight by converting the qualitative
values gleaned from the literature search. Once all links on the map had been fully
articulated with the rankings of each of the 93 linguistic terms (three for each link),
we refined the virtual FCM (shown in Figure 4) by substituting quantitative values for the previous qualitative values (see Figure
8).

Results

Experimentation with the weighted Fuzzy Cognitive Map

Experimentation with the weighted FCM was conducted, (see Algorithm 1), to ensure
that it would perform as expected and that the map had captured the dynamics of the
factors which affect levels of homelessness. We applied as the transformation function f of Equation 2. This choice is made as we are interested in understanding the impact
of increase or decrease of initial concept values on the overall stability of the
map [91].

Prototypical scenarios, similar to those used for the simplified FCM (Figure 3), were constructed with the goal of finding the extreme case most likely to result
in homelessness, the extreme case least likely to result in homelessness and several
middle cases, more closely representing the possibilities present in the real world,
where levels of homelessness are less certain.

The output of each prototypical case was interpreted through knowledge gleaned during
the literature search/scan and the opinion of the criminologist-researcher on the
team. Each example case had a variety of concepts activated at varying levels. The
models were then permitted to iterate as necessary to reach a stable state (no further
movement, positive or negative, for all concepts in the model). Final iterations are
reported for each model.

•Case 1: Most likely to result in homelessness. The concepts of addiction, family breakdown, government assistance, and mental illness
were activated at levels considered sufficiently high to dominate the system leading
to certain homelessness as shown in Table 5. It has been empirically determined that these concepts are often found together
and often precede homelessness [52,70,83]. Addiction and mental illness are often co-morbid and both commonly precede family
breakdown [51]. During times of increased addiction and mental illness in society it is the usual
reaction of the government to put into place policies and funding which will address
these problems [93].

•Tracking the effect of these concepts at strengths set to approximately 0.50, the
graph initially shows that government assistance is at a lower rate and then sharply
rises to address the increasing levels of addiction, mental illness, and family breakdown
in the modeled society. However, it takes little time before the triple threat of
addiction, mental illness and family breakdown overwhelm the system and levels of
homelessness rise dramatically where they remain at a steady, high rate (indicated
by the flat line at the top of the graph, Figure 9).

•Case 2: Least likely to result in homelessness. The concepts of addiction, education, income, family breakdown, and social network
support were activated at levels considered sufficiently high to dominate the system
leading to a certain outcome of no homelessness as shown in Table 6. In this case, the protective factors of education, income, and social network support
protect society from the negative effects of addiction and prevent homelessness. The
link between higher levels of education and higher levels of income have been well
documented [72]. Given that education prepares individuals to think creatively and to problem-solve,
it is surmised that those with higher levels of education would have a greater ability
to negotiate the complex rules that often are associated with government assistance.
Those who are wealthy and educated are also much more likely to be capable of identifying
and acquiring the services they might need, such as being able to pay for family counseling
rather than being wait-listed for government supplied family counseling.

•From Figure 10, it is noted that this model shows a initial dip in levels of income and education
in the first iterations as society attempts to deal with the addictions and threat
to family cohesion that result from the addictions. However, very quickly, the protective
factors of income, education, and social network support overwhelm the negative factors
and the threat of homelessness diminishes and remains at levels close to zero (as
indicated by the flat line at the bottom of Figure 10). Over time, the threat of family breakdown is also eliminated and income and education
both rise back to their initial levels.

•This second model demonstrates the critical importance of factors such as income
- which lead to health, acquisition of knowledge, better food and health care; and
education - which lead to wealth and all the positive factors which wealth can purchase.
Though addictions are shown as present in this modeled society, the low levels are
unable to overwhelm the model. Through model testing it became apparent that levels
of addiction lower than 0.30 often fail to overwhelm the positive factors, as long
as social support and education are both present at fairly high levels, see Figure
10. Much of the empirical literature support this [41,59,78]. Those with high levels of social support such as family, church, social groups,
community groups, school friends and community friends are often better able to weather
threats such as addictions and family breakdown.

•Case 3: Uncertain outcome of homelessness. In this model, we activated low levels of addiction and social network, high levels
of education and income, and moderate levels of family breakdown as shown in Table
7. In this case, the protective factors of education and income delay the onset of
homelessness but are insufficiently strong to prevent rising levels as the model iterates.
Over time, due to family breakdown and the diminishing social network support, addictions
begin to rise and as addictions rise, the likelihood of homelessness rapidly increases.
This model demonstrates, once again, the importance of family and social support as
well as the incredibly negative effects of drug addiction, both as a cause and result
of family breakdown.

•As in the case of the common-sense map of homelessness (Figure 3), this final model (Figure 11), acted in a manner which was fully explainable based on information acquired during
the literature search and prior knowledge of the research team. This allowed for confidence
that the model was functioning as it ought to and that we had captured not only a
number of the integral aspects which contribute to homelessness, but that they were
functioning in the direction and strengths which approximated real-life conditions.

Figure 11.Activated concepts at levels most closely representing a typical real-world case with
graphical representation of the impact of concepts on levels of homelessness over
time.

Analysis of network concepts

The purpose of this network analysis is to compare the degree of impact each of the
concepts exerts on the model. During network analysis, we varied the initial value
of a single concept from 0.1 to 1 while keeping the initial values of all other concepts
at a static level; except for the concept representing homelessness. After several
iterations, the value of homelessness was recorded. Then, for each factor, a plot
of the value of homelessness versus the initial value of the concept was recorded.
Ideally, for a factor with a positive effect on homelessness, the value of homelessness
should increase as the value of the factor increases, gradually converging to a positive
value. Concepts which have the reverse - a negative effect on homelessness, should
demonstrate a decrease in homelessness as they are increased. Concepts which have
higher convergent rates should demonstrate a greater impact on levels of homelessness.

To conduct the network analysis we first set the initial values for all concepts at
a level of 0.5 and checked the levels of homelessness after 5 iterations. At this
level and number of iterations, the majority of the plots resulted in a straight line
at a value of +1. This told us that the initial value of the factor (0.1 to 1) made
no difference on levels of homelessness and, obviously, was no help to our analysis.
After analyzing the map, we tried reducing the level of the initial values for all
concepts as well as reducing the number of iterations. Through a gradual reduction
process we found that by setting the initial concept values at 0.01 and running three
iterations we were able to generate reasonable and useful plots (see Figure 12) which could then be compared for effects on levels of homelessness.

Figure 12.Comparison of the affects of individual concepts on levels of homelessness (a) shows
the impact of Addiction, Criminal Justice System, Cost of housing and Social Network
on Homelessness (b) highlights the impact of Education, Family Breakdown, Government
Assistance and Income on Homelessness and (c) depicts the impact of Mental Illness,
NGO, Poverty and Childhood hardships.

Plots can be examined in pairs or groupings so that the effect of the concepts on
levels of homelessness can be compared for both intensity and speed. For example,
in comparing the plots for, “Addictions”, and, “Cost of Housing”, it can be seen that
they both are monotonically increasing. However, the plot for “Addictions” demonstrates
a more dramatic increase, resulting in a quicker convergence to +1 than does the plot
for “Cost of Housing”. Therefore it can be concluded that addictions have a greater
impact on homelessness than does cost of housing.

Another way to visually analyze the impact of various factors on homelessness is through
box plota (see Figure 13). Making the same comparison, “Addictions” to “Cost of Housing”, it can be seen that
the plot of “Addictions” has a narrower median and longer lower quantile. The size
of the box determines the variability of concepts, for instance, the size of the box
of “Cost of Housing” is greater than size of the box of “Addictions” indicating that
the impact of housing cost is more variable and hence not a strong indicator [94].

Figure 13.Boxplot comparison of the affects of individual concepts on levels of homelessness.

Measure of centrality

Another approach to analyze the most influential factor is through measures of centrality. There are also other measurements for analyzing an FCM, but here we focus on this
property. In this subsection, we describe the results of the analysis based on two
types of centrality: degree centrality and closeness centrality. Degree centrality
of each node/concept, in a given weighted and directed graph, is defined as the sum
of the absolute values of the weights of the outgoing and incoming edges [8,95]. For the node, x, of the graph G=<V,E> the degree centrality is mathematically defined as:

(4)

where wxy and wyx are the weights of the edge from x to y and the edge from y to x, respectively. Degree centrality of a graph indicates how strongly a concept node
in a FCM affects other concept nodes of the graph [96].

Closeness centrality of a node is the inverse of the sum of the lengths of the shortest
paths between that node and all other nodes. For the node, x, of the graph G=<V,E>, the closeness centrality is mathematically defined as:

(5)

where dxy denotes the length of the shortest path from node x to node y. Closeness centrality indicates how quickly a concept node affects other nodes of
the FCM [96].

Note: For closeness centrality the distance measured between each pair of nodes is the
inverse of the weight of the corresponding edge in the FCM. If there is no edge between
nodes then the distance from the one node to the second node would obviously be infinite.
Since the FCM is not strongly connected, the length of the shortest path for some
pairs of nodes is, in fact, infinite. This then causes the closeness centrality for
that node to drop to zero. For example, the length of shortest path for each node
to the node, “Cost of Housing”, is infinite. This makes the centrality of all nodes
to be zero. To conquer this problem, we choose a numerical value which is large enough
to be considered as an infinite value. Since the distance measure between each pair
of nodes is defined as the inverse of the weight between the nodes of the FCM, the
greatest distance between each two nodes would be 4. This value is corresponding to
the edge between “poverty” and “addiction”, whose weight is 0.25. The FCM has 14 concepts,
thus each path of the FCM will, at most, have 13 edges. Therefore, the length of each
path will be at most 4×13=52, which is still an overestimation of the paths in the
graph. Regarding this value, we picked 100 as an large enough value. This approach
is similar to the Big-M method described in operation research theories [97]. Please note that changing 100 to a greater value, may change nodes’ closeness centrality,
but the order of the nodes’ closeness centrality will not change.

The result of the degree and closeness centrality computation in our FCM is displayed
in Table 8. As shown, the concept “Education” has the greatest degree centrality while the concept
“Cost of Housing” has the least. This means that “Education” gives and receives the
greatest direct influence on all other concepts, whereas “Cost of Housing” gives and
receives the least. Closeness centrality was determined to act similarly to degree
centrality in that “Education” has the greatest amount of degree centrality whereas
“Cost of Housing” has the least. This means that “Education” exerts the greatest force
on the map in reference to closeness centrality with changes in “Education” resulting
in the most prominent changes in the other concepts. Likewise, changes in “Cost of
Housing” would result in the least amount of change in all other concepts. These results
are consistent with the results of the overall experiment.

Discussion

This study demonstrates the efficacy of using FCM to graphically represent and simulate
the actions and interactions present in the social, personal, and structural factors
related to homelessness. The FCM is particularly suited to modeling this type of problem
due to its ability to incorporate vast amounts of information, synthesizing what is
known about a problem and then allowing for meaningful simulations. The FCM is particularly
suitable due to its dynamic nature and ability to simulate potential policy changes
and show predicted outcomes on levels of homelessness. Further, the FCM helps to identify
those factors that exert the greatest impact in a complex system, in this case: affordable/appropriate
housing, access to social support services for those with addictions/mental illness,
family support for those with children, positive community support and rental supplements.

The problem of homelessness is really situated in factors that occur at the micro-,
meso-, and macro-levels of society; future research should aim to refine the FCM by
sorting factors into their appropriate levels thereby allowing differentiation between
what the individual is potentially capable of controlling and that which he or she
is not. This would allow for clearer identification of where government policy changes
would have the greatest effect. Future refinements must also capture the effect of
time. Many factors affect the system differently as time progresses (i.e., unemployment
insurance) and this would help to make the system more closely replicate reality.
Future maps may also wish to include factors which affect the system but which did
not make it into this one such as early brain injury in childhood, sexual/physical/emotional
abuse in childhood, and learning disabilities - all of which have been shown to affect
levels of homelessness.

The initial construction of this map demonstrated the disparity between the empirical
truth of homelessness and what the researchers had learned over a lifetime of media
and social propaganda. This has implications for government policy-making and, again,
demonstrates the usefulness of FCMs for describing complex social problems such as
homelessness.

Conclusion

The FCM built to model the complex social system of homelessness reasonably represented
reality for the sample scenarios. This provided evidence that FCMs are a viable alternative
for conceptualizing homelessness and that a literature search of peer reviewed, academic
literature is a reasonable foundation upon which to build the model. Further, it was
determined that the direction and strength of relationship between concepts included
in this map are a reasonable approximation of their action in reality. However, the
concept, homelessness, in this study, is used as a consequent variable. In reality, many of the concepts
including homelessness concept could be an antecedent concept resulting in more complex loops. The flexibility
of limiting the complexity is one of the advantages of constructing and using FCMs
for social science research.

Dynamic modeling does, however, have it’s limitations and this work should be regarded
as purely exploratory. For one, by basing our concepts off of peer reviewed literature
that was searched semi-systematically there is a possibility of not capturing all
possible terms. Future work should search for papers and terms in a similar fashion
as systematic or scoping reviews where inclusion and exclusion criteria are highly
scrutinized and analyzed by several research team members. A second limitation concerns
the interpretation of the results from the FCM. FCMs, and dynamic models more broadly,
have the luxury of experimenting with problems in an environment that is encapsulated
from the real-world. It should be noted that every societal issue carries with it
its own contextual element that cannot always be captured by a modeling environment.
Further, FCMs do not fully replicate the mirco-level interactions that may prove to
be powerful in determining meso- and macro-level outcomes. Future work should aim
to incorporate these influences in to their models and interpretations as best possible.
Lastly, dynamic models are exploratory and we can not reasonably assume that outcomes
presented in this research will be realized in the real world.

This research provides empirical support for the usefulness of this model, not only
for researchers and social scientists, but for others who reside within a society
where homelessness is experienced. This model is based on a limited collection of
published, peer reviewed scholarly articles but despite this limitation, does justify
the use of FCM techniques as a useful tool to analyze the complex situation of homelessness.
The role of FCM for the purpose of modelling complex social systems has been strongly
supported by this research and should continue to be utilized in future studies.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

VKM and VD conceived the idea and formulated mathematical model. TW, SN, PG, RC and
VKM implemented the computational model. HKM, CF wrote the paper along with VKM. All
authors critically analyzed the simulations, reviewed the manuscript, read and approved
the final version.

Acknowledgements

Initial work on this research project was conducted during the IRMACS Modelling Summer
School. This research was supported by the SFU CTEF MoCSSy program. We are also grateful
for technical support from the IRMACS Centre, Simon Fraser University, BC.

References

Strand R: Helping the homeless: the struggle between community values and political ideologies.

Obiedat M, Samarasinghea S, Strickert G: A New Method for Identifying the Central Nodes in Fuzzy Cognitive Maps using Consensus
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